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Chapter 8: Problem 5

Ist \(V\) einer der R盲ume \(C_{c}(X), C_{0}(X), C_{b}(X), C(X)\) und \(I: V \rightarrow \mathbb{K}\) eine positive Linearform, so gilt \(|I(f)| \leq I(|f|)\) f眉r alle \(f \in V\).

Short Answer

Expert verified
Yes, the inequality \(|I(f)| \leq I(|f|)\) holds true for any \(f \in V\), provided that \(I: V\rightarrow \mathbb{K}\) is a positive Linearform.

Step by step solution

01

Understanding the problem

First, it's important to know function spaces \(C_{c}(X)\), \(C_{0}(X)\), \(C_{b}(X)\), \(C(X)\) and their properties, and also understand what a positive Linearform is. In this context, \(I: V\rightarrow \mathbb{K}\) is a positive Linearform, which means it's a linear function mapping \(V\) to scalars and it's positive for all \(f \in V\) that are positive.
02

The inequality

Now, focus on the inequality \(|I(f)| \leq I(|f|)\). As per the definition of absolute values, we know that for any real number \(x\), \(|x| \leq a\) if and only if \(-a \leq x \leq a\). We need to show that \(-I(|f|) \leq I(f) \leq I(|f|)\). Let's consider only the left part of this inequality first. Since \(I\) is a positive linear form, it is known that \(I(-f) \leq 0\) for any \(f\) that is positive. Hence, \(I(-|f|) \leq 0\) which simply rewrites to \(-I(|f|) \leq 0\). Now, we also know that \(0 \leq I(f)\) for positive \(f\) because \(I\) is positive. Put together, we find \(-I(|f|) \leq I(f)\).
03

Proving the second part of the inequality

Now let's consider the second part of this inequality, \(I(f) \leq I(|f|)\). We know that \(f \leq |f|\) as per the definition of absolute values. It is also known that if \(a \leq b\), then \(I(a) \leq I(b)\), since \(I\) is positive. Applying this principle to \(f \leq |f|\), we find that \(I(f) \leq I(|f|)\). Thus, we have proved this part of the inequality as well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear form
In mathematics, a linear form is a linear map from a vector space to its field of scalars. Consider a vector space, denoted as V, over a field \(\mathbb{K}\). A linear form is a function \(I: V \rightarrow \mathbb{K}\). For any vectors \(u, v \in V\) and scalars \(a, b \in \mathbb{K}\), the linear form satisfies these key properties:
  • Additivity: \(I(u+v) = I(u) + I(v)\)
  • Homogeneity: \(I(av) = aI(v)\)
These properties ensure the function \(I\) preserves the structure of the space. In measure theory and analysis, understanding linear forms is fundamental because they help in mapping functions from spaces like \(C(X)\) or \(C_0(X)\) to scalars, facilitating function evaluation and integration.
Function spaces
Function spaces are collections of functions that share a common property, often equipped with a particular structure. In this context, we consider four types of function spaces:
  • \(C_c(X)\): space of continuous functions with compact support.
  • \(C_0(X)\): space of continuous functions vanishing at infinity.
  • \(C_b(X)\): space of bounded continuous functions.
  • \(C(X)\): space of all continuous functions on \(X\).
Each of these spaces serves different purposes in analysis. They work with properties of continuity, boundedness, and control over a limit. For example, functions in \(C_0(X)\) are useful when considering limits as they ensure that functions approach zero at infinity, ideal for integration over infinite domains. Understanding function spaces helps comprehend how different constraints on the functions can lead to varying types of function analysis.
Positive linear form
A positive linear form is a special type of linear form where the function preserves positivity under application. For a linear form \(I: V \rightarrow \mathbb{K}\) to be positive, it must satisfy:
  • If \(f \in V\) and \(f \geq 0\), then \(I(f) \geq 0\).
This property ensures that applying a positive linear form to non-negative elements of the function space results in non-negative scalars. These forms are integral in various fields such as functional analysis and measure theory because they guarantee the alignment of orientation and magnitude in calculations. This alignment is particularly crucial when stating and proving inequalities, as we see with the inequality in analysis discussed in this context.
Inequality in analysis
Inequalities are powerful tools in analysis, providing limits and bounds for expressions. In this specific exercise, the inequality \(|I(f)| \leq I(|f|)\) characterizes limits within the function space. Here's why this inequality holds:
  • By the properties of absolute value, we have for any function \(f\), \(|f|\) is always greater than or equal to \(f\).
  • The positivity of the linear form \(I\) ensures that \(I(f)\) will not exceed \(I(|f|)\), given \(f \leq |f|\).
This inequality demonstrates consistent behavior that guarantees a form of control or approximation in the behavior of functions across the space. In measure theory, such inequalities establish necessary conditions for integration theories and help in defining measures and their properties effectively, facilitating precise and meaningful calculations.

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Most popular questions from this chapter

Jede positive Linearform \(I: C_{b}(X) \rightarrow \mathbb{K}\) ist stetig bez. der Supremumsnorm.

Es seien \(X\) ein vollst盲ndig regul盲rer Hausdorff-Raum, \(\beta X\) die Stone- Cech-Kompaktifizierung von \(X\), und f眉r \(f \in C_{b}(X)\) sei \(\hat{f}: \beta X \rightarrow \mathbb{K}\) die eindeutig bestimmte stetige Fortsetzung von \(f\) auf \(\beta X\) (s. Aufgabe 2.7). Jeder positiven Linearform \(I: C_{b}(X) \rightarrow \mathbb{K}\) entspricht verm枚ge \(\hat{I}(h):=I(h \mid X) \quad(h \in C(\beta X))\) eine positive Linearform \(\hat{I}: C(\beta X) \rightarrow \mathbb{K}\), und zu dieser Linearform \(\hat{I}\) geh枚rt nach Darstellungssatz \(2.5\) genau ein Radon-Ma脽 \(\hat{\mu}: \mathfrak{B}(\beta X) \rightarrow[0, \infty[\) mit $$ \hat{I}(h)=\int_{\beta X} h d \hat{\mu} \quad(h \in C(\beta X)) $$ Daher wird \(I\) gem盲脽 $$ I(f)=\int_{\beta X} \hat{f} d \hat{\mu} \quad\left(f \in C_{b}(X)\right) $$ durch ein Radon-Ma脽 \(\hat{\mu}\) auf \(\mathfrak{B}(\beta X)\) beschrieben. Umgekehrt entspricht jedem Radon-Ma脽 \(\hat{\mu}\) auf \(\mathfrak{B}(\beta X)\) verm枚ge (*) eine positive Linearform \(I: C_{b}(X) \rightarrow \mathbb{K}\) - Es sei nun \(X\) sogar lokalkompakt. Dann ist \(X\) eine offene Teilmenge von \(\beta X\) (s. z.B. ENGELKING [1], S. 221, Theorem 3.5.8). Ferner seien \(I: C_{b}(X) \rightarrow \mathbb{K}\) eine positive Linearform und \(\mu\) das nach \((2.22),(2.23)\) zugeh枚rige endliche Radon-Ma脽 auf \(\mathfrak{B} .\) Zeigen Sie: \(I\) wird genau dann durch \(\mu\) dargestellt gem盲脽 \((2.25)\), wenn \(\hat{\mu}((\beta X) \backslash X)=0\), und dann ist \(\mu=\hat{\mu} \mid \mathfrak{B}(X)\).

Es sei \(\mu: \mathfrak{A} \rightarrow[0, \infty]\) ein Ma脽, wobei \(\mathfrak{A} \supset \mathfrak{B}\) eine \(\sigma\)-Algebra ist. a) Ist \(\left(A_{n}\right)_{n \geq 1}\) eine Folge von au脽en (bzw. innen) regul盲rer Mengen aus \(\mathfrak{A}\), so ist \(\bigcup_{n=1}^{\infty} A_{n}\) von au脽en (bzw. innen) regul盲r. b) Ist \(\left(A_{n}\right)_{n \geq 1}\) eine Folge von au脽en (bzw. innen) regul盲rer Mengen aus \(\mathfrak{A}\) mit \(\mu\left(A_{n}\right)<\) \(\infty(n \in \mathrm{N})\), so ist \(\bigcap_{n=1}^{\infty} A_{n}\) von au脽en (bzw. innen) regul盲r.

Es sei \(X\) ein Hausdorff-Raum, und f眉r \(a \in X\) sei \(\delta_{a}(B):=1\), falls \(a \in B\), und \(\delta_{a}(B):=0\), falls \(a \notin B \quad(B \in \mathfrak{B})\). Sind dann \(a_{1}, \ldots, a_{n} \in X\) paarweise verschieden und \(\lambda_{1}, \ldots, \lambda_{n}>0\) so ist \(\mu:=\lambda_{1} \delta_{a_{1}}+\ldots+\lambda_{n} \delta_{a_{n}}\) ein Radon-Maf mit \(\operatorname{Tr} \mu=\left\\{a_{1}, \ldots, a_{n}\right\\} .\) Ist umgekehrt \(\mu\) ein Radon-Ma脽 mit \(\operatorname{Tr} \mu=\left\\{a_{1}, \ldots, a_{n}\right\\} \quad\left(a_{1}, \ldots, a_{n}\right.\) paarweise verschieden), so gibt es \(\lambda_{1}, \ldots, \lambda_{n}>0\), so da脽 \(\mu=\lambda_{1} \delta_{a_{1}}+\ldots+\lambda_{n} \delta_{a_{n}}\)

Es seien \(X\) ein vollst盲ndig regul盲rer Hausdorff-Raum und \(\mu\) ein Radon-Ma脽 auf \(\mathfrak{B}\). a) F眉r \(a \in X\) gilt \(a \in \operatorname{Tr} \mu\) genau dann, wenn f眉r alle \(f \in C^{+}(X)\) mit \(f(a)>0\) gilt \(\int_{X} f d \mu>0\). b) Ist \(U \in \mathfrak{O}\), so gilt \(\mu(U)=0\) genau dann, wenn f眉r alle \(f \in C^{+}(X)\) mit \(f \mid U^{c}=0\) gilt \(\int_{X} f d \mu=0\)
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